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\title{Math 2001 \\ Equivalence relations}

\begin{document}
\maketitle



\vfill

\begin{hw*} Book exercises: Due Wednesday, April 13.

Section 11.0: 5, 7.

Section 11.1: 2, 8.

Section 11.2: 2, 4, 8.

Section 11.3: 4.
\end{hw*}

\noindent
\textbf{Proofs.}

Friday, April 8: first draft of Proof 13.

Monday, April 11: final draft of Proof 12.

Wednesday, April 13: final draft of Proof 13 and first draft of Proof 14.

\HR

\begin{example}
Let $A = \{1,2,3,4,5,6\}$, and let $R$ be an equivalence relation on $A$ defined by
\begin{align*}
R = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (2,3), (3,2), (4,6), (6,4), (4,5), (5,4), (5,6), (6,5)\}.
\end{align*}
List the equivalence classes of $R$.

\vspace{2in}

\end{example}

\begin{example}
Let $R$ be an equivalence relation on $A$, where $A = \{a,b,c,d,e\}$. Suppose that $(a,d) \in R$ and $(b,c) \in R$. Write out the elements of $R$, and draw the graph of $R$.
\end{example}

\vspace{2in}

\begin{example}
Let $R$ be the relation on $\bbz$ defined by
\begin{align*}
R = \{(a,b) : a,b \in \bbz, 3a-5b \text{ is even}\}.
\end{align*}
Describe the equivalence classes.
\end{example}

\pagebreak

\begin{thm}
Suppose that $R$ is an equivalence relation on a set $A$, and suppose also that $a,b \in A$. Then $[a] = [b]$ if and only if $(a,b) \in R$.
\end{thm}

\begin{proof}
{\color{white}.}
\vspace{2in}
{\color{white}.}

\end{proof}

\begin{defn}
A \emph{partition} of a set $A$ is a set of non-empty subsets of $A$ such that the union of all the subsets is equal to $A$, and the sets are pairwise disjoint. That is, if $X$ and $Y$ are in the partition, then $X \cap Y = \varnothing$.
\end{defn}

\begin{example}
Find all the partitions of $A = \{a,b,c\}$.
\end{example}

\vspace{1in}

\begin{thm}
Let $R$ be an equivalence relation on $A$. Then $\{[a] : a \in A\}$ is a partition of $A$.
\end{thm}

\end{document}